Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Renormalization methods in KAM theory

It is well known that an integrable (in the sense of Arnold-Jost) Hamiltonian system gives rise to quasi-periodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the Kolmogorow-Arnold-Moser (KAM) theorem states that, provided some non-degeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasi-periodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space.

Bayesian Inference for Retrospective Population Genetics Models Using Markov Chain Monte Carlo Methods

Genetics, the science of heredity and variation in living organisms, has a central role in medicine, in breeding crops and livestock, and in studying fundamental topics of biological sciences such as evolution and cell functioning. Currently the field of genetics is under a rapid development because of the recent advances in technologies by which molecular data can be obtained from living organisms. In order that most information from such data can be extracted, the analyses need to be carried out using statistical models that are tailored to take account of the particular genetic processes. In this thesis we formulate and analyze Bayesian models for genetic marker data of contemporary individuals. The major focus is on the modeling of the unobserved recent ancestry of the sampled individuals (say, for tens of generations or so), which is carried out by using explicit probabilistic reconstructions of the pedigree structures accompanied by the gene flows at the marker loci. For such a recent history, the recombination process is the major genetic force that shapes the genomes of the individuals, and it is included in the model by assuming that the recombination fractions between the adjacent markers are known. The posterior distribution of the unobserved history of the individuals is studied conditionally on the observed marker data by using a Markov chain Monte Carlo algorithm (MCMC). The example analyses consider estimation of the population structure, relatedness structure (both at the level of whole genomes as well as at each marker separately), and haplotype configurations. For situations where the pedigree structure is partially known, an algorithm to create an initial state for the MCMC algorithm is given. Furthermore, the thesis includes an extension of the model for the recent genetic history to situations where also a quantitative phenotype has been measured from the contemporary individuals. In that case the goal is to identify positions on the genome that affect the observed phenotypic values. This task is carried out within the Bayesian framework, where the number and the relative effects of the quantitative trait loci are treated as random variables whose posterior distribution is studied conditionally on the observed genetic and phenotypic data. In addition, the thesis contains an extension of a widely-used haplotyping method, the PHASE algorithm, to settings where genetic material from several individuals has been pooled together, and the allele frequencies of each pool are determined in a single genotyping.

Radiative transfer, interception and scattering in coniferous forests : models and applications for production ecology and remote sensing

This work develops methods to account for shoot structure in models of coniferous canopy radiative transfer. Shoot structure, as it varies along the light gradient inside canopy, affects the efficiency of light interception per unit needle area, foliage biomass, or foliage nitrogen. The clumping of needles in the shoot volume also causes a notable amount of multiple scattering of light within coniferous shoots. The effect of shoot structure on light interception is treated in the context of canopy level photosynthesis and resource use models, and the phenomenon of within-shoot multiple scattering in the context of physical canopy reflectance models for remote sensing purposes. Light interception. A method for estimating the amount of PAR (Photosynthetically Active Radiation) intercepted by a conifer shoot is presented. The method combines modelling of the directional distribution of radiation above canopy, fish-eye photographs taken at shoot locations to measure canopy gap fraction, and geometrical measurements of shoot orientation and structure. Data on light availability, shoot and needle structure and nitrogen content has been collected from canopies of Pacific silver fir (Abies amabilis (Dougl.) Forbes) and Norway spruce (Picea abies (L.) Karst.). Shoot structure acclimated to light gradient inside canopy so that more shaded shoots have better light interception efficiency. Light interception efficiency of shoots varied about two-fold per needle area, about four-fold per needle dry mass, and about five-fold per nitrogen content. Comparison of fertilized and control stands of Norway spruce indicated that light interception efficiency is not greatly affected by fertilization. Light scattering. Structure of coniferous shoots gives rise to multiple scattering of light between the needles of the shoot. Using geometric models of shoots, multiple scattering was studied by photon tracing simulations. Based on simulation results, the dependence of the scattering coefficient of shoot from the scattering coefficient of needles is shown to follow a simple one-parameter model. The single parameter, termed the recollision probability, describes the level of clumping of the needles in the shoot, is wavelength independent, and can be connected to previously used clumping indices. By using the recollision probability to correct for the within-shoot multiple scattering, canopy radiative transfer models which have used leaves as basic elements can use shoots as basic elements, and thus be applied for coniferous forests. Preliminary testing of this approach seems to explain, at least partially, why coniferous forests appear darker than broadleaved forests in satellite data.

Methods to improve gene signal: Application to cDNA microarrays

Microarrays are high throughput biological assays that allow the screening of thousands of genes for their expression. The main idea behind microarrays is to compute for each gene a unique signal that is directly proportional to the quantity of mRNA that was hybridized on the chip. A large number of steps and errors associated with each step make the generated expression signal noisy. As a result, microarray data need to be carefully pre-processed before their analysis can be assumed to lead to reliable and biologically relevant conclusions. This thesis focuses on developing methods for improving gene signal and further utilizing this improved signal for higher level analysis. To achieve this, first, approaches for designing microarray experiments using various optimality criteria, considering both biological and technical replicates, are described. A carefully designed experiment leads to signal with low noise, as the effect of unwanted variations is minimized and the precision of the estimates of the parameters of interest are maximized. Second, a system for improving the gene signal by using three scans at varying scanner sensitivities is developed. A novel Bayesian latent intensity model is then applied on these three sets of expression values, corresponding to the three scans, to estimate the suitably calibrated true signal of genes. Third, a novel image segmentation approach that segregates the fluorescent signal from the undesired noise is developed using an additional dye, SYBR green RNA II. This technique helped in identifying signal only with respect to the hybridized DNA, and signal corresponding to dust, scratch, spilling of dye, and other noises, are avoided. Fourth, an integrated statistical model is developed, where signal correction, systematic array effects, dye effects, and differential expression, are modelled jointly as opposed to a sequential application of several methods of analysis. The methods described in here have been tested only for cDNA microarrays, but can also, with some modifications, be applied to other high-throughput technologies. Keywords: High-throughput technology, microarray, cDNA, multiple scans, Bayesian hierarchical models, image analysis, experimental design, MCMC, WinBUGS.

Computational Methods for Reconstruction and Analysis of Genome-Scale Metabolic Networks

Metabolism is the cellular subsystem responsible for generation of energy from nutrients and production of building blocks for larger macromolecules. Computational and statistical modeling of metabolism is vital to many disciplines including bioengineering, the study of diseases, drug target identification, and understanding the evolution of metabolism. In this thesis, we propose efficient computational methods for metabolic modeling. The techniques presented are targeted particularly at the analysis of large metabolic models encompassing the whole metabolism of one or several organisms. We concentrate on three major themes of metabolic modeling: metabolic pathway analysis, metabolic reconstruction and the study of evolution of metabolism. In the first part of this thesis, we study metabolic pathway analysis. We propose a novel modeling framework called gapless modeling to study biochemically viable metabolic networks and pathways. In addition, we investigate the utilization of atom-level information on metabolism to improve the quality of pathway analyses. We describe efficient algorithms for discovering both gapless and atom-level metabolic pathways, and conduct experiments with large-scale metabolic networks. The presented gapless approach offers a compromise in terms of complexity and feasibility between the previous graph-theoretic and stoichiometric approaches to metabolic modeling. Gapless pathway analysis shows that microbial metabolic networks are not as robust to random damage as suggested by previous studies. Furthermore the amino acid biosynthesis pathways of the fungal species Trichoderma reesei discovered from atom-level data are shown to closely correspond to those of Saccharomyces cerevisiae. In the second part, we propose computational methods for metabolic reconstruction in the gapless modeling framework. We study the task of reconstructing a metabolic network that does not suffer from connectivity problems. Such problems often limit the usability of reconstructed models, and typically require a significant amount of manual postprocessing. We formulate gapless metabolic reconstruction as an optimization problem and propose an efficient divide-and-conquer strategy to solve it with real-world instances. We also describe computational techniques for solving problems stemming from ambiguities in metabolite naming. These techniques have been implemented in a web-based sofware ReMatch intended for reconstruction of models for 13C metabolic flux analysis. In the third part, we extend our scope from single to multiple metabolic networks and propose an algorithm for inferring gapless metabolic networks of ancestral species from phylogenetic data. Experimenting with 16 fungal species, we show that the method is able to generate results that are easily interpretable and that provide hypotheses about the evolution of metabolism.

Computational Methods for Detecting Large-Scale Chromosome Rearrangements in SNP Data

Large-scale chromosome rearrangements such as copy number variants (CNVs) and inversions encompass a considerable proportion of the genetic variation between human individuals. In a number of cases, they have been closely linked with various inheritable diseases. Single-nucleotide polymorphisms (SNPs) are another large part of the genetic variance between individuals. They are also typically abundant and their measuring is straightforward and cheap. This thesis presents computational means of using SNPs to detect the presence of inversions and deletions, a particular variety of CNVs. Technically, the inversion-detection algorithm detects the suppressed recombination rate between inverted and non-inverted haplotype populations whereas the deletion-detection algorithm uses the EM-algorithm to estimate the haplotype frequencies of a window with and without a deletion haplotype. As a contribution to population biology, a coalescent simulator for simulating inversion polymorphisms has been developed. Coalescent simulation is a backward-in-time method of modelling population ancestry. Technically, the simulator also models multiple crossovers by using the Counting model as the chiasma interference model. Finally, this thesis includes an experimental section. The aforementioned methods were tested on synthetic data to evaluate their power and specificity. They were also applied to the HapMap Phase II and Phase III data sets, yielding a number of candidates for previously unknown inversions, deletions and also correctly detecting known such rearrangements.

Statistical and Information-Theoretic Methods for Data-Analysis

In this Thesis, we develop theory and methods for computational data analysis. The problems in data analysis are approached from three perspectives: statistical learning theory, the Bayesian framework, and the information-theoretic minimum description length (MDL) principle. Contributions in statistical learning theory address the possibility of generalization to unseen cases, and regression analysis with partially observed data with an application to mobile device positioning. In the second part of the Thesis, we discuss so called Bayesian network classifiers, and show that they are closely related to logistic regression models. In the final part, we apply the MDL principle to tracing the history of old manuscripts, and to noise reduction in digital signals.

Methods for Answer Extraction in Textual Question Answering

In this thesis we present and evaluate two pattern matching based methods for answer extraction in textual question answering systems. A textual question answering system is a system that seeks answers to natural language questions from unstructured text. Textual question answering systems are an important research problem because as the amount of natural language text in digital format grows all the time, the need for novel methods for pinpointing important knowledge from the vast textual databases becomes more and more urgent. We concentrate on developing methods for the automatic creation of answer extraction patterns. A new type of extraction pattern is developed also. The pattern matching based approach chosen is interesting because of its language and application independence. The answer extraction methods are developed in the framework of our own question answering system. Publicly available datasets in English are used as training and evaluation data for the methods. The techniques developed are based on the well known methods of sequence alignment and hierarchical clustering. The similarity metric used is based on edit distance. The main conclusions of the research are that answer extraction patterns consisting of the most important words of the question and of the following information extracted from the answer context: plain words, part-of-speech tags, punctuation marks and capitalization patterns, can be used in the answer extraction module of a question answering system. This type of patterns and the two new methods for generating answer extraction patterns provide average results when compared to those produced by other systems using the same dataset. However, most answer extraction methods in the question answering systems tested with the same dataset are both hand crafted and based on a system-specific and fine-grained question classification. The the new methods developed in this thesis require no manual creation of answer extraction patterns. As a source of knowledge, they require a dataset of sample questions and answers, as well as a set of text documents that contain answers to most of the questions. The question classification used in the training data is a standard one and provided already in the publicly available data.

Computationally Efficient Methods for MDL-Optimal Density Estimation and Data Clustering

The Minimum Description Length (MDL) principle is a general, well-founded theoretical formalization of statistical modeling. The most important notion of MDL is the stochastic complexity, which can be interpreted as the shortest description length of a given sample of data relative to a model class. The exact definition of the stochastic complexity has gone through several evolutionary steps. The latest instantation is based on the so-called Normalized Maximum Likelihood (NML) distribution which has been shown to possess several important theoretical properties. However, the applications of this modern version of the MDL have been quite rare because of computational complexity problems, i.e., for discrete data, the definition of NML involves an exponential sum, and in the case of continuous data, a multi-dimensional integral usually infeasible to evaluate or even approximate accurately. In this doctoral dissertation, we present mathematical techniques for computing NML efficiently for some model families involving discrete data. We also show how these techniques can be used to apply MDL in two practical applications: histogram density estimation and clustering of multi-dimensional data.

Matrix Decomposition Methods for Data Mining : Computational Complexity and Algorithms

Matrix decompositions, where a given matrix is represented as a product of two other matrices, are regularly used in data mining. Most matrix decompositions have their roots in linear algebra, but the needs of data mining are not always those of linear algebra. In data mining one needs to have results that are interpretable -- and what is considered interpretable in data mining can be very different to what is considered interpretable in linear algebra. --- The purpose of this thesis is to study matrix decompositions that directly address the issue of interpretability. An example is a decomposition of binary matrices where the factor matrices are assumed to be binary and the matrix multiplication is Boolean. The restriction to binary factor matrices increases interpretability -- factor matrices are of the same type as the original matrix -- and allows the use of Boolean matrix multiplication, which is often more intuitive than normal matrix multiplication with binary matrices. Also several other decomposition methods are described, and the computational complexity of computing them is studied together with the hardness of approximating the related optimization problems. Based on these studies, algorithms for constructing the decompositions are proposed. Constructing the decompositions turns out to be computationally hard, and the proposed algorithms are mostly based on various heuristics. Nevertheless, the algorithms are shown to be capable of finding good results in empirical experiments conducted with both synthetic and real-world data.